Management accounting--decision management: the learning curve equation has a number of applications in the manufacturing sector. Fortunately, the formula itself is fairly straightforward to learn for paper P2.

The principle underlying learning curves is generally well
understood: if we perform tasks of a repetitive nature, the time we take
to complete subsequent tasks reduces until it can reduce no more. This
is relevant to management accounting in the two key areas of cost
estimation and standard costing.

Before we look at these we need to understand the maths. Imagine
that we have collected the following information for the production of
eight units of a product: it takes 1,000 hours to produce the first
unit; 600 hours to produce the second unit; 960 hours to produce the
third and fourth units; and 1,536 hours to produce the remaining four
units. There is clearly a learning curve effect here, as the production
time per unit is reducing from the initial 1,000 hours.

Learning curves are initially concerned with the relationship
between cumulative quantities and cumulative average times (total
cumulative time divided by cumulative quantity). The relationship in
this case is shown in table 1. Notice that, as the cumulative quantity
doubles, the cumulative average time reduces by 20 per cent. In other
words, subsequent cumulative average times can be obtained by
multiplying the previous cumulative average time by 80 per cent. This is
an example of an 80 per cent learning curve. A learning curve is
addressed in percentage terms, depending upon the relationship between
the cumulative average times when the cumulative quantities are
doubling. For example, if the cumulative average time were 1,000 hours
at the production of the first unit, 700 hours at the production of the
second, 490 hours at the fourth, 343 hours at the eighth and so on, this
would be a 70 per cent learning curve.

The learning curve formula is needed when dealing with situations
that do not fit into this doubling-up pattern.

A learning curve is geometric with the general form Y = a[X.sup.b].

Y = cumulative average time per unit or batch.

a = time taken to produce initial quantity.

X = the cumulative units of production or, if in batches, the
cumulative number of batches.

The first batch of a new product has just been made. The batch size
was 20 units and the total time taken was 200 hours--ie, an average of
10 hours per unit. A 90 per cent learning curve is expected to apply.
You are required to estimate the following:

a The cumulative average time for the first two batches.

b The total time to produce 40 units.

c The incremental time for 41 to 60 units--ie, a third batch of 20
units.

The solutions are as follows:

a This is a batch situation, so the Y value will be the cumulative
average time per batch for two batches of 20 units.

a = 20 x 10 = 200 hours (ie, the time for the first batch).

b = log 0.9 / log 2 = -0.152.

X = 40 / 20 = 2 (ie, two cumulative batches).

Y = 200([2.sup.-0.152]) = 180 hours per batch.

We could have avoided using the formula here and simply multiplied 200 by 0.9, because it was a doubling-up situation.

b A total time is needed so, as in example one, all we need to do
is multiply Y by X:

2 batches x 180 hours per batch = 360 hours.

c The incremental time for the third batch equals the total time
for 60 units minus the total time for 40 units.

We already have the total time for 40, so we need the total time
for 60 (ie, X = 3):

Y = 200([3.sup.-0.152]) = 169.24 hours per batch.

Total time for 60 units = 169.24 x 3 = 507.72 hours.

Incremental time = 507.72 - 360 = 147.72 hours.

Now that we've looked at the mechanics, let's consider a
couple of examples of how the learning curve can be applied.

Example three: cost estimation

BB plc uses a marginal costing system. You have been asked to
provide calculations of total variable costs for a contract for one of
its products, based on the following alternative situations:

1 A contract for one order of 600 units.

2 Contracts for a sequence of individual orders of 200, 100, 100
and 200 units. Four separate costings are required.

It's expected that the average unit variable cost data for an
initial batch of 200 units will be as follows:

2 For sequential individual orders of 200, 100, 100 and 200 units,
four separate costings are required. The costing for the first batch of
200 will be straightforward as we will simply use the average unit cost
data initially given.

So the total variable cost for the first batch of 200 units is:

Direct material: 200 x 120 = 24,000 [pounds sterling]

Direct labour:

Department A: 200 x 64 = 12,800 [pounds sterling]

Department B: 200 x 1,000 = 200,000 [pounds sterling]

Variable overhead: 0.25 x 212,800 = 53,200 [pounds sterling]

Total: 290,000 [pounds sterling]

For the next three batch orders we need to work out the incremental
costs. Direct material cost is constant per unit, so it's not a
problem. The variable overhead is 25 per cent of labour cost. Direct
labour cost is affected by learning curves, so we need to calculate the
incremental times using the learning curve formulas and then convert to
cost (see tables 2 and 3). Then we can complete the costings for the
second, third and fourth orders (see table 4).

Example four: standard costing

SC plc is establishing a revised standard cost for one of its
products. The product was introduced at the start of 2004 when the
standard variable cost for the first unit was as follows:

* Direct material: 10kg at 2 [pounds sterling] per kg.

* Direct labour: 10 hours at 8 [pounds sterling] per hour.

* Variable overhead: 10 hours at 4 [pounds sterling] per hour.

* Total variable cost per unit: 140 [pounds sterling].

During the year a 90 per cent learning curve was observed. The
cumulative production at the end of the third quarter was 50 units. The
budgeted production for the fourth quarter is 10 units.

You are required to calculate the following:

1 The standard cost per unit for the fourth quarter, assuming that
the 90 per cent curve is still appropriate.

2 The standard cost per unit for the fourth quarter, assuming that
peak efficiency was reached after the 50th unit was produced--ie, the
learning curve had reached a steady state.

The calculations are as follows:

The curve: Y = 10([X.sup.-0.152]).We need the incremental time for
the 10 units produced in the fourth quarter--ie, the total time for 60
units minus the total time for 50 units. The total time for 60 is the
cumulative average time x 60: 10([60.sup.-0.152]) x 60 = 322.2.

The total time for 50 is the cumulative average time x 50:
10([50.sup.-0.152]) x 50 = 276.

The incremental time per unit for the 10 units, therefore, is:
(322.2--276) + 10 = 4.62 hours per unit.

2 We are assuming that peak efficiency is reached after the 50th
unit, so we need the time it took to produce the 50th unit (ie, the
total time to produce 50 units minus the total time to produce 49
units).

Total time for 50 units: 276 hours (from solution 1).

Total time for 49 units: cumulative average time x 49:
10([49.sup.-0.152]) x 49 = 271.2 hours.

The incremental time to produce the 50th unit is: 276--271.2 = 4.8
hours.

Bill Brookfield ACMA is a freelance lecturer specialising in
management and financial accounting. He has worked for the London School
of Accountancy, the Emile Woolf College and the Financial Training
Company. One of his current commitments is co-ordinating and lecturing
on CIMA courses in Kazakhstan.

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